CARMA Research Group
Mathematical Modelling and Industrial Applications
Leader: Mike Meylan
The mathematical modelling and industrial applications group works on the application of mathematics to solving real-world problems. We have particular expertise in the implementation of differential equations to modelling complex problems in fluid dynamics and other processes. Our research can be broadly grouped into the following areas.
1) Mathematical Modelling. We have wide-ranging experience in modelling including extensive applications in fluid flows, wave scattering, material science, and biological processes including interactions between cells and swarming of insects.
2) Nanostructures and their applications in nanotechnology, includeing modelling electrorheologicalfluids, the mechanics of carbon nanostructures, nanomaterials used in biology and medicine and protein and other polymer chain structures using the calculus of variations.
3) Electromaterials in energy applications such as solar cells, energy storage and carbon capture.
4) Finite element methods and numerical solution of complex equations, including Mixed and Hybrid Finite Element Methods, Domain Decomposition Methods, Non-conforming Discretization Techniques, Nearly Incompressible Elasticity, Approximation Theory, Subset Selection & Variational Methods in Image Processing. The group also specialises in computaional techniques such as Dissipative Particle Dynamics (DPD), Molecular Dynamics (MD) and Smoothed Particle Hydrodynamics (SPH) which are used to model a complex system involving micro and nano materials.
Our group also focuses on working with industry to develop mathematical models to solve real industrial-based problems. The major upcoming event for our group is the Mathematics in Industry Study Group (MISG) for 2020-2022.
SR202, SR Building
"Optimization Methods for Inverse Problems from Imaging"
Prof Ke Chen
Optimization is often viewed as an active and yet mature research field. However the recent and rapid development in the emerging field of Imaging Sciences has provided a very rich source of new problems as well as big challenges for optimization. Such problems having typically non-smooth and non-convex functionals demand urgent and major improvements on traditional solution methods suitable for convex and differentiable functionals.
This talk presents a limited review of a set of Imaging Models which are investigated by the Liverpool group as well as other groups, out of the huge literature of related works. We start with image restoration models regularised by the total variation and high order regularizers. We then show some results from image registration to align a pair of images which may be in single-modality or multimodality with the latter very much non-trivial. Next we review the variational models for image segmentation. Finally we show some recent attempts to extend our image registration models from more traditional optimization to the Deep Learning framework.
Joint work with recent and current collaborators including D P Zhang, A Theljani, M Roberts, J P Zhang, A Jumaat, T Thompson.